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Properties

Label 16-1045e8-1.1-c0e8-0-2
Degree $16$
Conductor $1.422\times 10^{24}$
Sign $1$
Analytic cond. $0.00547251$
Root an. cond. $0.722165$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s − 2·5-s + 2·9-s + 2·11-s + 16-s − 2·19-s − 4·20-s + 25-s + 4·36-s + 4·44-s − 4·45-s − 2·49-s − 4·55-s − 6·61-s − 4·76-s − 2·80-s + 81-s + 4·95-s + 4·99-s + 2·100-s + 4·101-s + 121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + ⋯
L(s)  = 1  + 2·4-s − 2·5-s + 2·9-s + 2·11-s + 16-s − 2·19-s − 4·20-s + 25-s + 4·36-s + 4·44-s − 4·45-s − 2·49-s − 4·55-s − 6·61-s − 4·76-s − 2·80-s + 81-s + 4·95-s + 4·99-s + 2·100-s + 4·101-s + 121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{8} \cdot 11^{8} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(0.00547251\)
Root analytic conductor: \(0.722165\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{8} \cdot 11^{8} \cdot 19^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.025366279\)
\(L(\frac12)\) \(\approx\) \(1.025366279\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
19 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
good2 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
3 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
13 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
23 \( ( 1 - T )^{8}( 1 + T )^{8} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
37 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
43 \( ( 1 - T )^{8}( 1 + T )^{8} \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
53 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
61 \( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
67 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
89 \( ( 1 - T )^{8}( 1 + T )^{8} \)
97 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.46084979734364782089507129783, −4.31805634857752440322283090780, −4.30330419189378730050378248913, −4.28919572304668309142122561397, −3.90959356959646672118857638629, −3.75297522989244152125670996512, −3.71374492694138558055417172810, −3.71153123224264895993674475358, −3.57075966303427258618753091771, −3.15883124590015138532929464011, −3.13035201726454912294421340304, −3.11688648341726386143399485041, −3.00527518065222213294863655595, −2.77123326125437657688199875683, −2.56314855092256693030614263218, −2.42039324512963751380082861884, −2.21343594037291468385900676210, −1.97159446593751785041650031403, −1.83058042912092744945749190543, −1.67847950852051510447841201220, −1.63417776765186976561676001238, −1.51129745340735722523562639872, −1.49648414052574090114986320280, −0.77610587510365188026763751402, −0.77320405199323260281190916169, 0.77320405199323260281190916169, 0.77610587510365188026763751402, 1.49648414052574090114986320280, 1.51129745340735722523562639872, 1.63417776765186976561676001238, 1.67847950852051510447841201220, 1.83058042912092744945749190543, 1.97159446593751785041650031403, 2.21343594037291468385900676210, 2.42039324512963751380082861884, 2.56314855092256693030614263218, 2.77123326125437657688199875683, 3.00527518065222213294863655595, 3.11688648341726386143399485041, 3.13035201726454912294421340304, 3.15883124590015138532929464011, 3.57075966303427258618753091771, 3.71153123224264895993674475358, 3.71374492694138558055417172810, 3.75297522989244152125670996512, 3.90959356959646672118857638629, 4.28919572304668309142122561397, 4.30330419189378730050378248913, 4.31805634857752440322283090780, 4.46084979734364782089507129783

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.